
theorem Th1:
  for X being set holds proj1 X in bool union union X &
  proj2 X in bool union union X
  proof
    let X be set;
    now
      thus proj1 X c= union union X
      proof
        let x be object;
        assume x in proj1 X;
        then ex y be object st [x,y] in X by XTUPLE_0:def 12;
        hence x in union union X by XTUPLE_0:6;
      end;
      thus proj2 X c= union union X
      proof
        let x be object;
        assume x in proj2 X;
        then ex y be object st [y,x] in X by XTUPLE_0:def 13;
        hence x in union union X by XTUPLE_0:7;
      end;
    end;
    hence thesis;
  end;
